Transformation Of Equations

Transforming a quadratic to one with reciprocal roots uses the relations between symmetric functions, a standard JEE Advanced manipulation. For x^2 - 4x + 1 = 0, Vieta gives alpha + beta = 4 and alpha beta = 1. The new roots 1/alpha and 1/beta have sum (alpha + beta)/(alpha beta) = 4/1 = 4 and product 1/(alpha beta) = 1/1 = 1. A quadratic with this sum and product is x^2 - (sum)x + (product) = x^2 - 4x + 1 = 0, identical to the original because the product of roots is 1, making the equation self-reciprocal. Option x^2 + 4x + 1 flips the sign of the sum. Option x^2 - x + 4 swaps the roles of sum and product. Option x^2 - 4x - 1 changes the product sign. Hence the new equation is x^2 - 4x + 1 = 0. Plausibility check: a palindromic quadratic with equal first and last coefficients is invariant under the reciprocal-root transformation, exactly as observed here. Multiplying complex numbers as a rotation-and-scaling action turns many trigonometric and geometric problems into simple arithmetic on moduli and arguments. This polar viewpoint underlies the derivation of De Moivre's theorem and the placement of roots of unity, so fluency with multiplying moduli while adding arguments is foundational for the harder rotation-based locus questions that follow in the syllabus.